Question

Cho A= 1/2.3+1/3.4+1/4.5+...+1/99.100. So sánh A với 1/2

Answer 1

\(A=\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+...+\frac{1}{99\cdot100}\)

\(A=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-...+\frac{1}{99}-\frac{1}{100}\)

\(A=\frac{1}{2}-\frac{1}{100}\)

\(\frac{1}{2}-\frac{1}{100}< \frac{1}{2}\)

\(\Rightarrow A< \frac{1}{2}\)

Answer 2

\(A=\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{99.100}\)

\(A=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{99}-\frac{1}{100}\)

\(A=\frac{1}{2}-\frac{1}{100}\)

\(\Leftrightarrow A< \frac{1}{2}\)

Answer 3

\(A=\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{99.100}\)

\(A=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{99}-\frac{1}{100}\)

\(A=\frac{1}{2}-\frac{1}{100}\)

\(A=\frac{49}{100}\)

Vậy A < 1/2.Vì 1/2 = 50/100 nên 49/100 > 50/100 nên A > 1/2

Answer 4

Ta có : \(A=\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{99.100}\)

\(\Rightarrow A=\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+...+\left(\frac{1}{99}-\frac{1}{100}\right)\)

\(\Rightarrow A=\frac{1}{2}-\frac{1}{100}\)

Ta thấy : \(\frac{1}{2}-\frac{1}{100}< \frac{1}{2}\)

\(\Rightarrow A< \frac{1}{2}\)

Answer 5

\(A=\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{99.100}\)

\(A=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{5}+...+\frac{1}{99}+\frac{1}{100}\)

\(A=\frac{1}{2}-\frac{1}{100}\)

\(\Leftrightarrow A< \frac{1}{2}\)